countable basis

in exactly one way (where ax∈F). We are implicitly assuming, without further comment, that the vector space V has been given a topological structure or normed structure in which the above infinite sum is absolutely convergent (so that it converges to v regardless of the order in which the terms are summed).

The archetypical example of a countable basis is the Fourier series of a function: every continuous real-valued periodic function f on the unit circle S1=ℝ/2⁢π can be written as a Fourier series

f⁢(x)=∑n=0∞an⁢cos⁡(n⁢x)+∑n=1∞bn⁢sin⁡(n⁢x)

in exactly one way.

Note: A countable basis is a countable set, but it is not usually a basis.